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Error Analysis of Jacobi Spectral Galerkin and Multi-Galerkin Methods for Weakly Singular Volterra Integral Equations

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Abstract

In this article, a Jacobi spectral Galerkin method is developed for weakly singular Volterra integral equations of the second kind. To obtain the superconvergence results, we transform the domain of integration of Volterra integral equation to the standard interval \([-1, 1]\) using variable transformation and function transformation. We obtain the convergence rates both in infinity and weighted \(L^2\)-norm. We prove that the Jacobi spectral iterated Galerkin method shows improvement over the Jacobi spectral Galerkin method. We improve these results further by considering the Jacobi spectral iterated multi-Galerkin method. Theoretical results are justified by the numerical results.

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Acknowledgements

The research work of Gnaneshwar Nelakanti was supported by the National Board for Higher Mathematics, India, research project: No. 02011/6/2019NBHM(R.P)/R&D II/1236 dated 28/1/2019.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721 302, India

    Kapil Kant & Gnaneshwar Nelakanti

  2. Department of Mathematics, Indian Institute of Technology Jodhpur, Jodhpur, 342 037, India

    Moumita Mandal

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  1. Kapil Kant
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  2. Moumita Mandal
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  3. Gnaneshwar Nelakanti
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Correspondence to Kapil Kant.

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Kant, K., Mandal, M. & Nelakanti, G. Error Analysis of Jacobi Spectral Galerkin and Multi-Galerkin Methods for Weakly Singular Volterra Integral Equations. Mediterr. J. Math. 17, 20 (2020). https://doi.org/10.1007/s00009-019-1462-3

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  • DOI: https://doi.org/10.1007/s00009-019-1462-3

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